Answer :
Sure! To find [tex]\( P(k) \)[/tex] using synthetic division, let's set up and solve the problem using the given values and polynomial.
1. Identify the Polynomial and [tex]\( k \)[/tex]:
- Polynomial: [tex]\( P(x) = x^2 + x - 30 \)[/tex]
- [tex]\( k = 5 \)[/tex]
2. List the Coefficients:
The coefficients of [tex]\( P(x) \)[/tex] are:
- 1 (for [tex]\( x^2 \)[/tex])
- 1 (for [tex]\( x \)[/tex])
- -30 (constant term)
3. Set Up Synthetic Division:
Start the synthetic division by setting up the following:
- Write 5 (the value of [tex]\( k \)[/tex]) outside the division symbol.
- Write the coefficients of the polynomial in order inside the division symbol.
The setup looks like this:
```
5 | 1 1 -30
|
----------------
```
4. Perform the Division:
- Bring down the first coefficient, which is 1.
- Multiply this coefficient by 5 (the value of [tex]\( k \)[/tex]) and write the result underneath the next coefficient.
- Add the result to the next coefficient.
- First Step:
- Bring down the 1:
```
5 | 1 1 -30
|
----------------
1
```
- Multiply: [tex]\( 1 \times 5 = 5 \)[/tex], write under the second coefficient:
```
5 | 1 1 -30
| 5
----------------
1
```
- Add: [tex]\( 1 + 5 = 6 \)[/tex], place under the line:
```
5 | 1 1 -30
| 5
----------------
1 6
```
- Second Step:
- Multiply the last result by 5: [tex]\( 6 \times 5 = 30 \)[/tex], write under the third coefficient:
```
5 | 1 1 -30
| 5 30
----------------
1 6
```
- Add: [tex]\( -30 + 30 = 0 \)[/tex]:
```
5 | 1 1 -30
| 5 30
----------------
1 6 0
```
5. Interpret the Results:
The numbers at the bottom row represent the coefficients of the quotient polynomial and the remainder. Here:
- The quotient polynomial is [tex]\( x + 6 \)[/tex] (represented by 1 and 6).
- The remainder is 0, which means [tex]\( P(5) = 0 \)[/tex].
So, the value of [tex]\( P(k) \)[/tex], where [tex]\( k = 5 \)[/tex], is [tex]\( 0 \)[/tex]. This means [tex]\( 5 \)[/tex] is a root of the polynomial [tex]\( P(x) = x^2 + x - 30 \)[/tex].
1. Identify the Polynomial and [tex]\( k \)[/tex]:
- Polynomial: [tex]\( P(x) = x^2 + x - 30 \)[/tex]
- [tex]\( k = 5 \)[/tex]
2. List the Coefficients:
The coefficients of [tex]\( P(x) \)[/tex] are:
- 1 (for [tex]\( x^2 \)[/tex])
- 1 (for [tex]\( x \)[/tex])
- -30 (constant term)
3. Set Up Synthetic Division:
Start the synthetic division by setting up the following:
- Write 5 (the value of [tex]\( k \)[/tex]) outside the division symbol.
- Write the coefficients of the polynomial in order inside the division symbol.
The setup looks like this:
```
5 | 1 1 -30
|
----------------
```
4. Perform the Division:
- Bring down the first coefficient, which is 1.
- Multiply this coefficient by 5 (the value of [tex]\( k \)[/tex]) and write the result underneath the next coefficient.
- Add the result to the next coefficient.
- First Step:
- Bring down the 1:
```
5 | 1 1 -30
|
----------------
1
```
- Multiply: [tex]\( 1 \times 5 = 5 \)[/tex], write under the second coefficient:
```
5 | 1 1 -30
| 5
----------------
1
```
- Add: [tex]\( 1 + 5 = 6 \)[/tex], place under the line:
```
5 | 1 1 -30
| 5
----------------
1 6
```
- Second Step:
- Multiply the last result by 5: [tex]\( 6 \times 5 = 30 \)[/tex], write under the third coefficient:
```
5 | 1 1 -30
| 5 30
----------------
1 6
```
- Add: [tex]\( -30 + 30 = 0 \)[/tex]:
```
5 | 1 1 -30
| 5 30
----------------
1 6 0
```
5. Interpret the Results:
The numbers at the bottom row represent the coefficients of the quotient polynomial and the remainder. Here:
- The quotient polynomial is [tex]\( x + 6 \)[/tex] (represented by 1 and 6).
- The remainder is 0, which means [tex]\( P(5) = 0 \)[/tex].
So, the value of [tex]\( P(k) \)[/tex], where [tex]\( k = 5 \)[/tex], is [tex]\( 0 \)[/tex]. This means [tex]\( 5 \)[/tex] is a root of the polynomial [tex]\( P(x) = x^2 + x - 30 \)[/tex].
